International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
3. VISIBILITY ANALYSIS OF TPM 
3.1 Visibility Relationship Between Prisms and Base 
Triangles 
In this paper, the following expression is used in this paper for 
prism A (or triangle A) that is hided by prism B (or triangle B) 
from 3-D view point O (or 2-D viewpoint O): 
Al» B8.:0 
And if prism A (or triangle A) is not hided by prism B (or 
triangle B), ie. A[^ B :O , the following expression is used in 
this paper: 
AP B:O 
B 
  
  
  
  
  
Figure 2. Visibility relationship from viewpoint O. 
The following theorem simplifies visibility relationship 
between triangular prisms in TPM. 
Theorem 1: For any pair of triangular prism A, B in TPM, if 
AD B:0 
then 
Bp 4:0 
The following theorem relates the visibility of triangle prisms 
from the viewpoint to visibility of base triangles. 
Tpr, 
  
  
  
  
  
Figure 3. Theorem 2. 
Theorem 2: Let Tpr be TPM which includes 
prisms (7pr,, Tpr, ,...,Tpr,) , the base triangle group of Tpr be 
Tri = {Tri,,Tri,,...,Tri,} , and O be the vertical projection of 
3-D viewpoint O, called 2-D viewpoint corresponding to O. 
For any pair of i and j (i * j), 
Tpr, I» Tpr, :0 = Trl Triz :o 
And as the contrapositive of the upper proposition, 
Tri, |p Tri, :0 => Tpr, P Tpr, :O 
3.2 Visibility Sorting 
3.2.1 Definition: The result of visibility sorting of triangle 
prisms from 3-D viewpoint O is a sequence 
S := (8$,8,,.., ,) in which any pair of prisms, 
> SO 
Similarly, a visibility sorting sequence of base triangles from 2- 
D viewpoint O is defined as a sequence s:-(5,,5,,.,5,) in 
which any pair of triangles, 
i« j — s $, 50 
Here we call a procedure to obtain a visibility sorting sequence 
“visibility sorting”. 
Theorem 2 leads to the following important theorem. 
Theorem 3: If ST7ri is a visibility sorting sequence of base 
triangles of a TPM, the sequence of prisms S7pr , where prisms 
corresponding to STri are arranged in the same order, also 
composes a visibility sorting sequence. 
3.2.2 Visibility Sorting of Triangles: Visibility sorting of 
triangles from 2-D viewpoint O can be executed in the 
- following steps. 
Step 1: Let the triangles be Tri := (Tri, Tri, ,..., Tri, j . Generate 
TIN with triangles edges and corners. If the viewpoint o is 
outside of triangles, include o for TIN generation. Let these 
triangles be Tri! = (Tri), Tri. Tri, } - 
Step 2: Choose one triangle which includes o. Let it be the first 
triangle S, in the sorted triangle sequence, and remove it from 
the TIN (see Figure 4 (1)). We call the polygon around removed 
area “front polygon”. Here the front polygon is 5S, itself. 
Step 3: Remove one triangle that is not hidden by other 
triangles from o . Add the triangle to the sorted triangle 
sequence. 
Step 4: Repeat Step 3 and remove until all triangles in TIN are 
removed (see Figure 4 (2)-(12)). 
Step 5: From the sorted triangle sequence, remove triangles not 
included in Tri:- (Tri, Tri,,...,Tri,) . The remaining sorted 
triangle sequence is the resultant visibility sorting sequence. 
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